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Diaspora of a Mathematics of Argument R. Loui Dept of Computer Science Washington University St. Louis Outline I. Intellectual History of Process-Based Models of Reasoning II. Some Technical Issues regarding Argument III. Foundations: A. Probability B. Decision C. Legal Reasoning D. Belief Revision/Deontic Logic E. Negotiation F. Rhetoric IV. Future Work A. Fairness B. Computation CS TALK Scope of My Current CS Work cgi in gawk book with S. Sachs independent co-malloc for localizing dynamically allocated objects optimal average hash chain length for gawk malloc with a vmstat time series estimator for elective memory expansion gnu release(s) with M. Waldvogel, M. Pachos, K. Krouse something with FPGA's patent & license with J. Lockwood, J. Moscola, M. Pachos CS TALK Scope of My Current CS Work (cont.) purely probabilistic negotiating agents model, simulations real-time object recognition for aerial targets hiding among non-combatants half-baked ideas, proposal with R. Pless AI and Law service journal, ICAIL, JURIX, special issues, workshops, JD-PhD's, no $ WHAT IS LOGIC? What do Computer Scientists think is Logic? Roughly: Hilbert-Russell-Whitehead tradition: 1. there is one correct logic: it is either the predicate logic or the propositional logic or both 2. entailment (syntactic or semantic?) has something to do with mathematical proof 3. logic codifies correct ways of reasoning 4. logic has something to do with the success of hardware WHAT IS LOGIC? What do Computer Scientists think is Logic? (cont.) Some more advanced members of our species: Knowledge Representation 1. logics are like programming languages; can be chosen or designed w/o metaphysical consequence 2. some logics are more expressive than others 3. some logics license more inferences than others 4. inferential license and expressiveness are complementary WHAT'S NEW/DIFFERENT What is the Difference between Argument and Deduction? Diagram of an argument <p, { < {a,b}, p >, <{c}, a>, <{d}, b> }> Diagram of a proof <p1, p2, …, p> Obvious Argument Deduction nondemonstrative demonstrative if p then defeasibly q if p then materially q nonmonotonic monotonic argument proof subargument subproof counterargument (counterproof?) defeat (fallibility? corrigibility?) inconsistency-tolerating inconsistency-degenerating Less Obvious Argument Deduction focus is on metalanguage: focus is on object language: conflict, rebuttal, warrant and, or, not anytime: ideal warrant w.r.t. arguments commitment at all t produced in time t Warranted(S,t) Thm(S), Proved(S,t) constructive: nonconstructive p's warrant underdetermined ampliative: nondeterminism nonampliative: conclusions of process in the meanings of words Less Obvious (continued) Argument Deduction strategy-based: constraint-based: choices proof = constraint-propagation dialectical unilateral sits between objectivist and invites principle of charity: relativist conceptions of truth formally, 10-20 years old formally, 100-150 years old 20thC seminal in social sciences 20thC dominant in logic Variations of Logic Modal Logic opaque contexts >> notation for beliefs about beliefs Fuzzy Logic predication weakened >> smoother control, washing machines Multivalued Logic truth weakened >> semantic curiosities, reductions Relevance Logic implication weakened >> model of limited inferential capacity Variations of Logic (cont.) Intuitionist Logic weak negation added >> first step toward elevation of process Counterfactual Logic second implication added >> plausible alternative conditional Paraconsistent Logic meaning from inconsistency >> proof-theory for consistent subsets Belief Revision recovery from inconsistency >> model of premise adoption/retraction Argument ties logic to computation in a fundamental way >> rewrite foundations of other fields INTELLECUTAL HISTORY 1. where did the idea of defeasibility come from? 2. where did the idea of procedural rationality come from? 3. where did the idea of argument come from? AI: "tweety is a bird, but tweety does not fly" McCarthy-Hayes -- Modal Belief 1973 Reiter --Closed World Databases 1978 Doyle -- TMS 1978 Kowalski -- PROLOG 1974 Clark -- Negation as Failure 1978 Argument systems 1987... Pollock -- Defeasible Reasoning 1968-1974-1986-1987 Nute -- Defeasible PROLOG 1985 Kyburg -- system for probability based on defeat 1961-1974 Epistemology: "it seems red therefore it is red" is defeasible Belzer -- Defeasible Reasoning 1986 Swain -- Epistemic Defeasibility 1978 Pollock -- Knowledge & Justification 1974 Sosa -- Conceptions of Knowledge 1970 Lehrer-Paxson -- Knowlege 1969 Firth -- Coherence 1964 Chisholm -- Perceiving 1957-1964 Ladd -- Structure of a Moral Code 1957 Reasoning (Qualitative Decision Theory): "doing a achieves the goal, therefore do a" is defeasible Nozick -- Practical Reason/Explanations 1981 Searle -- Prima Facie Obligations/Practical Reason 1978-1985 Raz -- Practical Reason/Norms 1970 Gauthier -- Practical Reasoning 1963 Ethical Reasoning: "a person has a prima facie obligation or responsibility" Glover -- Responsibility 1970 Nozick -- Moral Structures 1968 Feinberg -- Action and Responsibility 1965 Wellman -- Language of Ethics 1961 Brandt -- Blameworthiness and obligation 1958 Melden -- Action/Rights 1956-1959 Mackie -- Responsibility and Language 1955 Hare -- Language of Morals 1952 (note: Stevenson 1938 and Ross 1930) Political Justification: Barry -- Political Argument 1965 Rawls -- Pure Procedural Justice 1958-1974 Dialectics/Rhetoric: Rescher -- Dialectics 1977 Perelman -- Justice and Argument 1963 Toulmin -- Uses of Argument 1958 Origins: Ladd Raz Gauthier Wellman Brandt/Melden/Hare Barry Rawls Perelman/Toulmin Hart Origins (continued): Hart -- Ascription of Responsibility 1948 Wisdom -- Gods 1945 Waismann -- Verifiability 1951 Austin -- Speech Acts 1947? Wittgenstein -- Remarks on Foundations of Mathematics 1935? Keynes -- Treatise on Probability 1908 thesis begins “Part of our knowledge we obtain direct; and part by argument.” Bentham -- Principles of Morals and Legislation 185? Defeasibility: When the student has learnt that in English law here are positive conditions required for the existence of a valid contract, - his understanding of the legal concept of a contract is still incomplete, ... For ... he has still to learn what can defeat a claim that there is a valid contract, even though all these conditions are satisfied. The student has still to learn what can follow on the word "unless", which should accompany the statement of these conditions. This characteristic of legal concepts is one for which no word exists in ordinary English ... but the law has a word which with some hesitation I borrow and extend: this is the word “defeasible”... (Hart vs. Aristoteliean Society, 1951, p. 152) Process: ... [Principia Mathematica] gives rise to questions about the relation in which ordinary reasoning stands to this ordered system, and in particular, as to the precise connection between the process of inference, in which the older logicians were principally interested, but which [Russell] ignores, and the relation of implication, on which his scheme depends. The gradual perfection of the formal treatment ... had been to empty [logic] of content and to reduce it more and more to mere dry bones, until finally it seemed to exclude ... most of the principles, usually deemed logical, of reasonable thought. (Keynes vs. Russell, Whitehead, Ramsey, 1908/1921/1973, p. 118, 1972, p. 243) Formal Inconsistency: WITTGENSTEIN: Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone-- Because the thing works like this: if a man says 'I am lying' we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? ... It does not matter. ... it is just a useless language-game, and why should anyone be excited? TURING: ... one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong. WITTGENSTEIN: Yes -- and more: nothing has been done wrong. ... where will the harm come? (Wittgenstein vs. Turing, 1939, Hodges, p. 154) RECENT TECHNICAL QUESTIONS 0. knee-jerk reaction (deductivists) Q. isn't p > r, p & q > ~r reducible to p & ~q --> r p & q --> ~r ? A. no. what can be concluded with just p? r. does that imply ~q? no. TECHNICAL I. old issues (Touretzky, Horty, Thomason, 1987) Q. skeptical vs. credulous A. skeptical Q. ambiguity-propagating vs. blocking A. depends whether undercut or rebut Q. syntactic specificity 1. (strict specificity) p>r p & q > ~r 2. (shortcut specificity) p > q; q > r p > ~r 3. (defeasible specificity) p>r q > ~r p>q TECHNICAL (continued) A. keep it simple (Nute, 1990) A. appeal to convention (Simari-Loui, 1992) A. give explicit ordering (Lin-Shoham, 1987, Vreeswijk, 1991) A. provide for meta-argument about defeat e.g., context-dependent defeat (Prakken-Sartor, 1995) [r1] p > q [r2] r > ~q s > [r1>r2] TECHNICAL (continued) II. principles versus rules (Loui-Norman, Hage, Verheij, 1995-2001) Q. Is there a formal difference between: 1. no vehicles in the park 2. parks should be peaceful A. Rationales: can recall rationales during dispute A. Principles can be weighed: free speech versus privacy TECHNICAL (continued) III. rules for fair dialectic (Loui, Gordon, Vreeswijk, Lodder, 1992-2001) Q. What is the exact procedural burden? 1. pro: argument 1 for p 2. con: argument 2 for ~p, and argument 2 defeats argument 1. a. should con have the burden of showing argument 2 defeats argument 1? or b. should the claim be presumable and subject to dispute? TECHNICAL (continued) Q. are normal default rules fair? p:q/q 1. pro: argument 1 for p 2. con: a. proof of ~q b. (it does not suffice to argue ~q?) Q. what is the penalty of failed attempts to rebut? 1. pro: argument 1 for p based on b and c, etc. 2. con: ~b and ~c. 3. pro: why ~b? 4-15. con: ... pro: ... con loses the subdispute over ~b. 16. con: nevertheless, ~c. A. rhetorical costs: HIGH. logical costs: NONE? TECHNICAL (continued) IV. rules extracted from cases Q. What is the structure of a precedent case? A. (Raz, 1970) abcdef/q A. (Rissland-Ashley, 1985-1990) a+ b+ c+ d- e- f- / q+ A. (Loui-Norman, 1992) argument 1(a,(b,c); q) argument 2(a,d,e; ~q) argument 3(e,f; ~d) argument 4(c; ~f) _______________ q TECHNICAL (continued) Q. What is the rule of the case? A. (Loui-Norman, 1995) sufficient premises of arguments in dialectical subtree with leafs that are pro arguments abcdef>q but no false specificity: abcf>q A. (Prakken-Sartor, Bench-Capon, 1992-2001) {Argument1, Argument3} > {Argument2, Argument4} TECHNICAL (continued) V. criteria for theory-formation when theories are defeasible? (Peczenik, McCarty, 1997-2001) Q. Given a set of cases: case 1 a b d e f j q case 2 a b d ~q ... case n b d ~r what is the "best-fitting" set of defeasible rules? 1. all cases predicted by rules 2. no error (so far this is a learning problem with no simplicity measure) 3. sets of rules restricted or justified by principles? FOUNDATIONS: I. Probability A probability calculation is an argument. A statistical argument is an argument. Reference Class: Reichenbach (1949): "use the narrowest reference class for which there are adequate statistics" Kyburg (1961,1974,1983): maximum in a partial order? dominance ~= defeat. Each "inference structure" permits an argument from a different sample class. Prob(A | B C D)? Sample from BCD: 5 A's/9 Sample from BC: 14 A's/20 what is the logic of combinatorial significance tests? FOUNDATIONS: I. Probability (cont.) <S1,[p1,q1]> and <S2,[p2,q2]> disagree iff not([p1,q1] in [p2,q2]) and not([p2,q2] in [p1,q1]) <S1,[p1,q1]> defeats <S2,[p2,q2]> iff they disagree and S1 strict subset of S2 also Pollock (1985, 1990) who uses defeat explicitly FOUNDATIONS: I. Probability (cont.) Protocols Shafer on Monte-Hall type probability "paradoxes" (1985) the probability argument is improved through knowledge of the protocol Neyman-Pearson tradition of crucial tests two crucial tests produce two competing statistical arguments? FOUNDATIONS: II. Decision Problem of Small Worlds Savage (1954, 1967): considering fresh and stale should not change the calculation based on good and rotten. But of course it does. So: a grand world which contains all detail. pseudomicrocosm vs. real microcosm. Shafer-Tversky (1988): framing problems constructive decision theory FOUNDATIONS: II. Decision (cont.) Loui (1990): u(s) given T(P1,s) m(P1) = 5 u(s) given T(P2,s) m(P2) = -4 u(s) given T(P1 & P2, s) m(P1 & P2) = 6; u(s) = 5-4+6 ("defeasibility" of linearity) Holds(P1&P2, s) > u(s) = 7 defeats Holds(P1,s) > u(s) = 5 (defeasibility of valuation) FOUNDATIONS: II. Decision (cont.) But s is a lottery: s = {r/p; t;(1-p)} Prob(p) = .5; u(r) = 10; u(t) = 0; so expected u(s) = .5(10) +.5(0) = 5 (defeasibility of outcome) Simon (1955-1967) substantive vs. procedural rationality, yes, but more importantly: decision is more like chess than constraint-propagation: heuristic valuation changes as search/computation proceeds a defeasible independence/substitution axiom? paradoxes of certainty, menu-dependence, framing effects based on description FOUNDATIONS: III. Legal Reasoning FOUNDATIONS: IV. Belief Revision/Deontic Logic contrary-to-duty imperatives (Chisholm, 1963, Nute etc., 1996) 1. O(p-->q), O(p-->r) and O(~p) are consistent. (there can be two expiations) 2. O(~p) entails O(p --> x) for any x. (all expiations are obliged) von Wright (1982): "It only means that, if the prohibition is violated, the coordinated Contrary-to-Duty imperatives require, for their satisfaction, that both q and that r come true. ... If ... the conjunction of the two ... is a logical impossibility ..., the legislator would presumably take steps to remove the conflict." "deontic obligation" is different from "technical obligation" FOUNDATIONS: IV. Belief Revision/Deontic Logic 3. O(A|B), O(~A|C) entails ~(B&C) Alchourron (1993): "a set of conditional general norms entails ... a non-tautological sentence ... iff it follows in the logic for normative propositions that the authority has inconsistently normed some action for some circumstance." If norms are defeasible rules, no such problems: 1. two different arguments for expiation. 2. material conditionals are NOT deontic conditionals. 3. the entailment is not a result for defeasible conditionals FOUNDATIONS: IV. Belief Revision/Deontic Logic AGM belief revision: choice and refinement, ampliativity if p then defeasibly q if p and r then defeasibly ~q p --> q in K0 p & ~r --> q in K0[p&r-->~q][r] Alchourron (1993): "It seems to me unquestionable that the main [conditions] are the formal representation of the revisions effectively performed by an agent and of his dispositions to revise." "The particular details of the revisions (and the choice functions) are never analyzed by a logician (as a logician)..." Yes, defeasible conditionals would inform choice functions, but invoke "possible confusion of logic and revision," hiding "conceptually weaker conclusions" in "quiet darkness." FOUNDATIONS: V. Negotiation Acceptability can be argued: Fisher-Ury-Patton (1981): Principled negotiation gives arguments for proposals. “why not open the window?” “I’m cold” “I have a sweater” Sycara (1988-1995), Loui-Moore (1993-1997), Parsons-Jennings etc. (1998-2001): case-based arguments from precedent settlements “that raise was acceptable to you last year” Utility can be argued: search can lift utilities at the proposed settlements FOUNDATIONS: V. Negotiation Utility can be argued: Instead of strategic form {a1, …, a4} X {b1, …, b5} with utilities Ua and Ub, suppose OPTa(x,y) and OPTb(x,y), a hard optimization problem for each agent with parameters determined by the agreement Ua(x,y) is a’s current best solution for OPTa(x,y) Ua and Ub are lifted at <p,q> which is the focus of dialogue or when joint-problem-solving Sunk cost-of-search arguments lead to settlement FOUNDATIONS: VI. Rhetoric/"Informal Logic" Future Work: Fairness (Procedural) Claim 1. Fairness depends on the computational abilities of agents (the known subspace S x T of the possible strategies S* x T*; thus, rules are changed when a strategy s+ is discovered for which all known responses t are inadequate. Claim 2. To be fair (just), the procedure must construct its output upon the right inputs, with adequate monotonicity and invariance properties; thus, the justification of social procedure resembles the proof of program correctness. (concatenation) ex-post asymmetry of position should be the result of fair (just) ex-ante asymmetry adjusted only by the procedure’s effects on elective inputs (strategic choices). Future Work: Fairness (cont.) Obsv (political scientists). the purposes of the procedure can limit the degree of stochastics, the maximum variation of outcome, and the permissible input types. Claim 3. procedures should be non-dictatorial (for every important different kind of outcome, e.g., victory/defeat, there is a strategy pair that would reach this outcome). (dominance is more interesting…) Obsv rule symmetry and equivalent initial position are prima facie fair (but sometimes there are good reasons for bias, e.g. plaintiff) Claim 4. Fairness can be inherited from class relationships among procedural types. Future Work: Computation Observations. Social procedures which regulate/distribute/construct distributions are games. Social programming is like distributed programming (quantify over strategy tuples). Building societies is like inventing algorithms for distributed decision-making. Argument games. Welfare distributions. Elections. Tournaments. Objective. I want computer science to be at the foundation of the study of social procedures. Obstacle. Game-playing is not considered computation (yet). Claim. Two people playing chess compute the outcome of the game. Why do you have trouble with this claim? But not modems or chess tournaments for charities. Future Work: Computation Paradigmatic computation: on a machine: but, long division by hand? causally connected: but, two people doing long division? deterministic: but, probabilistic algorithms? locus of control: but, distributed algorithms? fully specified: but, pseudo-code? uncompiled code? algorithmic: but, protocols? interactions? (control systems?) non-elective: but, frequently arbitrary choices (e.g., search algorithms) Protocol-design is a kind of programming. Future Work: Computation Broad computation: intentional and teleological rule-following upon symbols. The existence of a program is the test for computation, not the existence of an algorithm. Not just any rule-following is computation, since the objects must be symbolic and the rule-following purposive and non-accidental.

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